Systems and methods for measuring relationships between investments and other variables

ABSTRACT

The systems and methods described herein can identify meaningful relationships between variables, such as particular investments or general asset classes. Unlike conventional correlation analysis, these systems and methods provide an improved technique of co-movement analysis that implements a threshold to eliminate data “noise” and then discretizes the remaining observations to normalize any outliers. Such co-movement analysis has numerous advantages over known techniques for characterizing relationships between variables.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/537,434, entitled “SYSTEMS AND METHODS FOR MEASURING RELATIONSHIPSBETWEEN INVESTMENTS AND OTHER VARIABLES,” filed Nov. 29, 2021, which isa continuation-in-part application of U.S. patent application Ser. No.17/226,874, entitled “SYSTEMS AND METHODS FOR MEASURING RELATIONSHIPSBETWEEN INVESTMENTS AND OTHER VARIABLES,” filed Apr. 9, 2021, which is acontinuation of U.S. patent application Ser. No. 15/948,962, entitled“SYSTEMS AND METHODS FOR MEASURING RELATIONSHIPS BETWEEN INVESTMENTS ANDOTHER VARIABLES,” filed Apr. 9, 2018, which is a continuation of U.S.patent application Ser. No. 14/015,257, entitled “SYSTEMS AND METHODSFOR MEASURING RELATIONSHIPS BETWEEN INVESTMENTS AND OTHER VARIABLES,”filed Aug. 30, 2013, which claims priority to U.S. Provisional PatentApplication Ser. No. 61/769,963, entitled “SYSTEMS AND METHODS FORMEASURING RELATIONSHIPS BETWEEN INVESTMENTS AND OTHER VARIABLES,” filedFeb. 27, 2013, each of which are incorporated by reference in theirentirety.

U.S. patent application Ser. No. 14/015,257 is also acontinuation-in-part of U.S. patent application Ser. No. 13/601,310,entitled “SYSTEMS AND METHODS FOR MANAGING INVESTMENTS,” filed Aug. 31,2012, each of which are incorporated by reference in their entirety.

U.S. patent application Ser. No. 14/015,257 is also acontinuation-in-part of U.S. patent application Ser. No. 13/601,386,entitled “SYSTEMS AND METHODS FOR MANAGING INVESTMENTS,” filed Aug. 31,2012, each of which are hereby incorporated by reference in theirentirety.

TECHNICAL FIELD

This invention relates generally to systems and methods for measuringand visualizing investments and other variables.

BACKGROUND

The primary objective of the investment management industry is tomaximize returns while minimizing risk. The process of assimilatingvarious investments into a portfolio that accomplishes this objective isone of the primary challenges for the industry. With the rise ofsophisticated investment strategies and products, the portfolioconstruction process only becomes more difficult as managers performanalysis across a wider variety of asset classes, sectors and marketsand attempt to quantify increasingly complex relationships. Whileconceptually sound techniques for optimal portfolio construction haveexisted for many years, the various assumptions underlying thesetechniques have not evolved with financial markets. Conventional toolsand statistics used in modern portfolio construction suffer from flawsin both assumptions and application. The tools incorrectly assume that asingle relational model (e.g., linear, curvilinear) or even multiplerelational models can define the complex and dynamic relationshipsbetween financial variables. In addition, practitioners usingconventional tools often prioritize statistical significance overeconomic significance. In doing so, practitioners prioritize the “fit”of a model over identifying potential relationships more important toprofit and loss. As a result, the financial industry has struggled toconstruct portfolios with optimum levels of risk and return.

Moreover, conventional tools do not provide a visual representation ofthe assets and how they relate to each other in a manner that is easy todigest for the viewer. In high-pressure and time-sensitive environments,where financial data changes rapidly, representing portfolio analysis ina manner that can be understood easily and quickly is highly desirable.

SUMMARY

In an attempt to better measure relationships between asset classes,sectors and markets, the systems and methods described herein establisha framework that can provide for portfolio construction with improvedlevels of risk and/or return. Analysts have conventionally relied oncorrelation models, but these statistics often fail to identifyimportant relationships or place too much emphasis on trivialrelationships. For example, a model based on correlation may be entirelyinsufficient when a long-term trend undergoes a sudden or even gradualchange.

The systems and methods described herein enable optimal portfolioconstruction based on a new relationship model providing numerousimprovements over conventional analysis, such as correlation. Further,the framework described herein allows for additional portfolio riskanalysis based on this new relationship model. The systems and methodscan identify previously hidden relationships between two or morevariables, further characterize known relationships between variables oreven reveal when there is no significant relationship between variables.The systems and methods described herein can also enable hedging complexderivative products and/or hybrid options (e.g., what is traded and/orembedded in longer-dated structured products). These products generallylean on covariance, which can under-state directionality and createunnecessarily large hedging costs.

The systems and methods described herein can inform hedging of complexderivative products and/or “hybrid options” (e.g. options that areexplicitly or implicitly contained in longer-dated structured products).Hedges for such products generally rely on covariance and therefore mayunderstate directionality and/or result in unnecessarily large hedgingcosts.

The systems and methods described herein have multiple applications inthe field of finance and investment management. For example, theframework can identify previously unknown relationships between assetclasses, sectors and markets. In some embodiments, the frameworkdisclosed herein can be used to analyze relationships between assetclasses in times of market stress, which are typically indicated bylarge price movements. For example, this framework can be used toidentify meaningful relationships that arise when an asset moves morethan a threshold amount (e.g., identifying which asset classes move morethan five percent when a general equity index moves more than fivepercent). In some embodiments, the systems and methods described hereincan enhance the application of mean-variance optimization in portfolioconstruction. Mean-variance portfolio optimization was developed byProfessor Harry Markowitz of San Diego, California, and this method iswidely used in the investment management industry for portfolioconstruction and management. The systems and methods described hereincan produce covariance measures that better model co-movement betweenfinancial variables, thereby improving mean-variance optimization.Furthermore, the systems and methods described herein, when applied tothe same input data, provide more forward-looking and robust measures ofexpected return and risk, thereby better identifying true risk-adjustedreturns. The framework described herein can also provide insight beyondthe particular variables under analysis, often revealing external trendsthat may affect those variables (e.g., buy-side trends in themarketplace).

The systems and methods described herein have additional applicationsoutside of finance and investment management. For example, the currentframework can be applied to sport statistics, behavioral statistics,employment statistics, real estate statistics, or any other measurableobjective data to identify relationships between variables. Moregenerally, the systems and methods described herein can be used in anyfield in which two or more variables behave according to a relationshipthat cannot be fully represented by existing analytical tools.

Additional features and advantages of various embodiments will be setforth in the description which follows, and in part will be apparentfrom the description. Other advantages will be realized and attained bythe structure particularly pointed out in the exemplary embodiments inthe written description and claims hereof as well as the appendeddrawings.

In one embodiment, a method comprises retrieving, by a server,performance data for a plurality of data records within an observationperiod; for at least one pair of data records within the plurality ofdata records, determining, by the server, whether a first data record ofa pair of data records and a second data record of the pair of datarecords have a positive union or a negative union based on each instancein which a respective value of the performance data for each data recordis above an upper threshold or below a lower threshold for the firstdata record or the second data record; displaying, by the server on agraphical user interface, a representation of the positive or negativeunion; and in response to receiving an indication of interaction withthe representation of the positive or negative union, dynamicallyrevising, by the server, the graphical user interface by displaying, forthe pair of data records, a visual indicator within four regions,wherein: a first region represents positive union with respect to theupper threshold and the lower threshold, a second region representsnegative union with respect to the upper threshold and the lowerthreshold, a third region represents positive union with respect to thelower threshold and negative union with respect to the upper threshold,and a fourth region represents negative union with respect to the lowerthreshold and positive union with respect to the upper threshold.

The method may also display a fifth region representing performance dataassociated with data records that are between the upper threshold andthe lower threshold.

In another embodiment, a computer system comprises a processor and anon-transitory computer-readable medium containing instructions thatwhen executed by the processor cause the processor to perform operationscomprising: retrieving performance data for a plurality of data recordswithin an observation period; for at least one pair of data recordswithin the plurality of data records, determining whether a first datarecord of a pair of data records and a second data record of the pair ofdata records have a positive union or a negative union based on eachinstance in which a respective value of the performance data for eachdata record is above an upper threshold or below a lower threshold forthe first data record or the second data record; displaying, on agraphical user interface, a representation of the positive or negativeunion; and in response to receiving an indication of interaction withthe representation of the positive or negative union, dynamicallyrevising the graphical user interface by displaying, for the pair ofdata records, a visual indicator within four regions, wherein: a firstregion represents positive union with respect to the upper threshold andthe lower threshold, a second region represents negative union withrespect to the upper threshold and the lower threshold, a third regionrepresents positive union with respect to the lower threshold andnegative union with respect to the upper threshold, and a fourth regionrepresents negative union with respect to the lower threshold andpositive union with respect to the upper threshold.

In another embodiment, a method comprises retrieving, by a server, atleast one of an observation period, an upper threshold, or a lowerthreshold from a template; for at least one pair of data records withina plurality of data records, determining, by the server, whether a firstdata record of a pair of data records and a second data record of thepair of data records have a positive union or a negative union based oneach instance in which a respective value of a performance data for eachdata record is above an upper threshold or below a lower threshold forthe first data record or the second data record; displaying, by theserver on a graphical user interface, for the pair of data records, avisual indicator within four regions, wherein: a first region representspositive union with respect to the upper threshold and the lowerthreshold, a second region represents negative union with respect to theupper threshold and the lower threshold, a third region representspositive union with respect to the lower threshold and negative unionwith respect to the upper threshold, and a fourth region representsnegative union with respect to the lower threshold and positive unionwith respect to the upper threshold.

In yet another embodiment, a method comprises retrieving, by a server,performance data for a plurality of data records within an observationperiod; for at least one pair of data records within a plurality of datarecords, determining, by the server, whether a first data record of apair of data records and a second data record of the pair of datarecords have a positive union or a negative union based on each instancein which a respective value of the performance data for each data recordis above an upper threshold or below a lower threshold for the firstdata record or the second data record; displaying, by the server on agraphical user interface, for the pair of data records, a visualindicator within four regions, wherein: a first region representspositive union with respect to the upper threshold and the lowerthreshold, a second region represents negative union with respect to theupper threshold and the lower threshold, a third region representspositive union with respect to the lower threshold and negative unionwith respect to the upper threshold, and a fourth region representsnegative union with respect to the lower threshold and positive unionwith respect to the upper threshold; and dynamically increasing, by theserver, a length of the observation period based on a confidence scorefor the positive or negative union.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and areintended to provide further explanation.

BRIEF DESCRIPTION OF THE DRAWINGS

The preferred embodiments of the present invention are illustrated byway of example and not limited to the following figures:

FIG. 1 depicts a system architecture, according to an embodiment.

embodiment.

FIG. 2 depicts a method of comparing two variables, according to an

FIG. 3 depicts a method of portfolio construction, according to anembodiment.

FIG. 4 depicts a method of portfolio construction, according to anembodiment.

FIGS. 5A-5B depict different methods used for portfolio construction andfor comparing two variables, according to an embodiment.

FIGS. 6-9 depict various graphical user interfaces displayed, accordingto an embodiment.

FIGS. 10 depicts a method of portfolio construction, analysis, andvisualization, according to an embodiment.

DETAILED DESCRIPTION

Various embodiments and aspects of the invention will be described withreference to details discussed below, and the accompanying drawings willillustrate the various embodiments. The following description anddrawings are illustrative of the invention and are not to be construedas limiting the invention. Numerous specific details are described toprovide a thorough understanding of various embodiments of the presentinvention. However, in certain instances, well-known or conventionaldetails are not described in order to provide a concise discussion ofembodiments of the present invention.

The embodiments described herein attempt to identify previously hiddenrelationships between two or more variables or further characterizeknown relationships between variables. This information has manyapplications in the field of finance and investment management. Forexample, information about the relationships between multiple variables(e.g., asset classes, deal codes, investment strategies, and/or sectorsor markets) can be used as an input during portfolio construction, suchas a measure of covariance across different variables. In anotherexample, when managing multiple investments, it may be useful to analyzerelationships between the investments to determine whether thoseinvestments are truly independent investments.

The current framework provides numerous advantages over known techniquesfor measuring relationships between variables. Such conventionaltechniques often rely on regression analysis, which can have severalshortcomings. Regression analysis, as used herein, may refer to thecommonly used ordinary least squares linear regressions encompassing anentire data population. For example, regression analysis typicallyrelies upon many data points to represent every movement of thevariables, but many of these movements may be minor changes that do notprovide any significant insight into the relationship between the twovariables. In other words, regression analysis often incorporates“noise” by including too many inconsequential data points. As anotherexample, regression analysis typically incorporates data points coveringevery movement over an extended period of time, but certainrelationships, such as those in financial markets, can changedrastically in relatively short periods of time. When such a changeoccurs, a large number of data points from the distant history mayimproperly weight the results and minimize the effect of a more recent,substantial movement. As another example, typical regression analysiscan rely on R² calculations, which use a straight line fit, butrelationships in the financial markets often do not follow straight linerelationships.

Generally, the systems and methods described herein can measure therelationship between variables by determining when the variables exceeda minimum absolute value change in the same or opposite directions. Therelationship between variables, as described herein, is known as the“Gerber relationship.” The Gerber relationship between two or morevariables (e.g., asset classes, sectors, or markets) is an alternativemeasure of co-movement between those variables. A Gerber relationshipbetween variables can be a positive relation (e.g., both variablesgenerally move in the same direction at the same time) or a negativerelation (e.g., both variables generally move in opposite directions atthe same time). A large positive relation may signify that the variablestypically move in the same direction, while a large negative relationmay signify that the variables typically move in opposite directions.

In contrast to conventional techniques, the systems and methodsdescribed herein can incorporate a threshold for filtering data pointsreflecting smaller variable movements that do not have any economicsignificance. In some embodiments, a threshold may be applied such thatthe Gerber relationship only considers data points reflecting a changegreater than a predetermined magnitude. For example, when measuring theGerber relationship between two asset classes, a threshold may beapplied such that relatively minor changes in the value of either assetclass can be filtered from the analysis. Any movement less than thethreshold may be considered “noise,” and filtering out those data pointsbelow the threshold may be desirable because they are likely toerroneously skew the analysis. After applying a threshold to filter outnoise, the remaining data points may be used to measure the Gerberrelationship between the variables. Accordingly, the Gerber relationshipcan overcome the problem of data noise caused by conventionaltechnique's over-inclusion of historical data in favor of moreimmediate, significant data about the variables.

In some embodiments, the systems and methods described herein may alsoapply a discretization process such that all data points exceeding thethreshold are given equal weight. For example, when measuring the Gerberrelationship between two asset classes, data points passing thethreshold may be discretized such that a modest movement barelyexceeding the threshold is given the same weight as a massive movementthat exceeds the threshold ten-fold. Any massive movement might beconventionally considered an outlier, and therefore, its magnitude couldhave been considered to erroneously skew an analysis. However, the eventof the massive movement may still be incorporated into this analysisbecause it has been discretized. In summary, measuring a Gerberrelationship can include implementing a threshold to eliminate datanoise and then discretizing the remaining observations to normalize anyoutliers while still incorporating these economically significantobservations into the analysis.

Generally, the systems and methods described herein can calculate astatistic quantifying the Gerber relationship between variables. Thisstatistic representing the Gerber relationship, as described herein, isknown as the “Gerber statistic.” In some embodiments, the Gerberstatistic can be a positive or negative number reflecting the relativedirection and strength of the relationship. Calculating the Gerberstatistic may include counting the number of instances when values ofboth variables changed beyond a threshold and considering whether thosechanges were both in the same direction or in opposite directions.Instances when both variables move beyond the threshold and in the samedirection (i.e., have a positive relation) are referred to herein as“positive unions,” while instances when both variables move beyond thethreshold and in opposite directions (i.e., have a negative relation)are referred to herein as “negative unions.” Only periods in which bothvariables have movements beyond the threshold may be considered whencalculating the Gerber statistic.

In some embodiments, a Gerber statistic can be a number between −100%and +100% that characterizes the Gerber relationship between a pair ofvariables. In one example of calculating a Gerber statistic, a thresholdvalue can be set at a predefined percentage value of the underlyingassets (e.g., 1%) for a period of 10 days. During those 10 days, a firstvariable and a second variable may have movements in the same oropposite directions. Each time period in which the movements of both ofthese variables exceed the threshold value can be compared to determinethe co-movement of the variables. One example method for calculating theGerber statistic can include determining the number of positive unionsminus the number of negative unions, all divided by a number of totalunions. Alternatively, the Gerber statistic can be calculated bydetermining the number of positive unions minus a number of negativeunions, all divided by the length of the period. The Gerber statistic isnot intended to be limited to any particular formula, but can includeany calculation of co-movement where a threshold is applied to eliminatenoise, and the remaining observations are compared for positive unions,negative unions, or both positive and negative unions.

In another example of calculating a Gerber statistic, there are 5 of the10 days where the value of the first variable moved more than thethreshold value. During those 5 days, the second variable only movedmore than the threshold value 4 times. Therefore, the number of totalunions is 4. During 3 of those 4 days, the first and second variablesmoved in the same direction (e.g., both positive or both negative), sothe number of positive unions is 3. During the 1 remaining day fromthose 4 days, the first and second variables moved in the oppositedirections (e.g., one positive and one negative), so the number ofnegative unions is 1. In this example, the Gerber statistic can becalculated as (3−1)/4, which is 50%. By implementing a threshold, anyinsignificant movements under the threshold value of $50,000 can beeliminated from the comparison. The remaining movements that exceed thethreshold are discretized. If one movement was $300,000 and anothermovement was $70,000, these amounts are considered movements above thethreshold value, but the magnitude above the threshold is not consideredpertinent to the measure. Each movement above the threshold value isgiven equal weight, so a value conventionally considered an “outlier”would not skew these results.

In some embodiments, a Gerber statistic near −100% may indicate that thetwo variables have a high negative Gerber relationship. In other words,when the two variables both experience large movements, they typicallymove in opposite directions. On the other hand, a Gerber statistic closeto 100% may indicate that the two variables have a high positiverelation. In other words, when the two variables both experience largemovements, they typically move in the same direction. Additionally, aGerber statistic around 0% may indicate that the two variables do nothave any movements beyond the threshold or a relatively equal number ofpositive and negative unions.

The systems and methods described herein can determine a Gerberrelationship and calculate a Gerber statistic. Upon identifyingvariables, the systems and methods can retrieve the appropriatehistorical data to measure the Gerber relationship and calculate theGerber statistic. As described herein, the systems and methods cancomprise a computer program embodied on a computer-readable medium thatcan automatically perform the functions described herein, retrieveinformation to perform these functions, and display or output theresults on a graphical user interface or provide the results to anothersystem for further processing.

In some configurations, the methods and systems described herein can beused to calculate relationships between financial variables in order toevaluate strategies in which the relationship between different assetreturns is critical to determining the probability of large loss. Inturn, the probability of large loss is critical in determiningappropriate investment leverage and/or the cost insuring against such aloss. These products include: (i) investments with open-ended losspotential but defined and non-recourse capital commitment; and (ii)specific cases of option replications involving multiple asset classes.The Gerber Statistic allows investors and intermediaries to bettermodel, visualize, interpret, and ultimately invest in such products.

In one example, consider an investment in a multi-strategy hedge fundwhich delivers consistently positive returns with high returns per unitof realized volatility, but low levels of absolute performance. Forexample, a fund could deliver 5% absolute return with 2.5% dailyannualized volatility employing a combination of ten differentunderlying strategies. Most investors would consider such returnsattractive due to the 2.0 Sharpe ratio (assuming interest rates at zerofor simplicity). However, these returns are less attractive in thecontext of earning sufficient absolute return on un-levered capital. Byemploying the methods and systems discussed herein (e.g., the GerberStatistic), the investor and/or intermediary can visualize the frequencyof times when the multi-strategy hedge fund would face losses greatercertain threshold on unlevered capital. More precisely, using such avisualization, market participants could determine the likelihood of thefund delivering returns below a threshold X% (most commonly −100%) withan inputted statistical confidence level (e.g., 99%). Such a calculationwould be more precise and relevant than a correlation analysis whichwould over-weight small upside moves relative to more important largedownside moves. Further, calculating a Gerber Statistic based on acombination of individual strategy returns is superior to merely lookingat the historic performance of the fund as fund allocations to differentstrategies are dynamic through time.

When using the methods and systems discussed herein, a computer systemcan allow an investor to select a degree of leverage to achieve a targetabsolute return while formally quantifying the frequency of large losseswhich would wipe out existing capital and require further commitment.Similarly, an intermediary could use the Gerber Statistic andstatistical confidence interval around the Gerber Statistic toappropriately price an insurance policy or put against such an event.With an insurance policy (e.g., put) in place, the multi-strategy hedgefund investment can offer sufficiently high levered returns whilefunctioning more like a “long only” allocation where the investor'smaximum loss is capital invested. We see multiple benefits of using theGerber Statistic when calculating risk and pricing puts (e.g.,insurance) on multi-strategy hedge fund investments. The end investoraccesses an investment which would be otherwise un-accessible orun-economic. In particular, retail investors could benefit from suchaccess, as they are often otherwise credit-constrained against takingleverage which could result in losses beyond initial capital committed.Said differently, the Gerber Statistic could expand the breadth of“retail structured products” to more complex strategies such asmulti-strategy hedge funds. Some institutional investors face similarconstraints and opportunities. The hedge fund itself benefits byaccumulating greater assets to invest which is one measure of successand profitability. An intermediary pricing the puts/insurance policyuses the Gerber Statistic to improve the pricing of its product,capturing more business in the process.

As a second example, an investor may desire to earn a payout if twoseparate events occur, such as the price of gold rising and the level ofinterest rates rising. Such an investment may be motivated by eitherspeculation or hedging purposes and is an increasingly commontransaction among many types of investors. In this example, a standardregression may prove sub-optimal because relationships may not be linearand/or all available data points may not be arranged in such a way as toproperly capture the complexity of such a payout. The relevant data toachieve the investor's goal may not just be the co-movement of the twoassets, but also the magnitude of movement and directional co-movementin periods when both assets are appreciating (e.g., a 9×9 matrix withcolumns defined as “up,” “flat,” and “down” for the level of interestrates and rows defined as “up,” “flat,” and “down” for the price ofgold). By employing a Gerber Statistic, investors and traders can modeland visualize such outcomes focused exclusively on the subset ofoutcomes where both assets are higher. Investors may care about both thefrequency of these outcomes relative to the entire population as well asthe magnitude of co-movement within that subset. Traders looking tohedge such a product could focus on implied probability distributions aspriced in the options markets. Investors looking to invest in such aproduct could compare these pricings to historical frequencies and/ortheir own forward-looking expectations.

In a non-investment application, consider an ice hockey team who winsgames by scoring more goals (offense) than they allow (defense).Focusing on defense and a simplified approach, allowing a goal can bedefined as a function of (i) the realized skill of defenseman 1 “D1,”(ii) the realized skill of defenseman 2 “D2,” and (iii) the realizedskill of the goaltender “G.” Given unlimited resources and availabletalent, a team could secure the best of each, paying for the bestavailable D1, D2, and G. Doing so would minimize goals allowed, but isnot realistic since teams face competition for players, limited budgetsfor paying players, and league-imposed limits on total salaries. Oneoption for the team is to divide its constrained budget evenly acrossthe three positions securing the best available player for each, wherebest is defined by some quantitative combination of qualitative scoutingreports and increasingly available precise player analytics. Selectingthe best available player for each position is analogous to atraditional linear regression approach securing the best possible teamacross the full universe of outcomes. By using a Gerber Statistic, theteam can instead visualize and focus on the subset of most relevantoutcome: goals allowed. By testing various combinations of D1, D2, andG, the team may find, for example that hiring an expensive G allows themto hire less expensive D1 and D2, freeing up salary for other positions.Further, the team could threshold this result to their specificcircumstances. For example, if they have a strong offense (or faceopponents with weak defense), the threshold may be set to minimizingfrequency of allowing more than three goals per game. If they have aweaker offense, the threshold may instead be set to two goals per game.

A further non-investment application of the methods and systemsdiscussed herein (Gerber Statistic) exists in the realm of healthcareand specifically for measurement and display of adverse patient outcomeswhere multiple treatments interact. For example, consider a patienttaking two medications to address two separate medical conditions: highcholesterol and a pain in the neck. Condition one is treated byanti-cholesterol medication. Condition two is treated by painsuppressing medication. Where the Gerber Statistic is valuable is inpresentation of this information. At present, the interactions betweenthe two medications are assumed to have already been studied andestablished. More clearly and consistently presenting this informationoffers three positive effects: (i) improving decision making forpatients who are trained in neither medicine nor statistics, (ii)providing more easily understandable metrics to doctors in high-pressuretime-sensitive situations, and (iii) motivating further broad and deepstudies of interactions creating data for future use. Using the methodsand systems described herein, a patient or healthcare provider couldvisit a website or mobile application which graphically displays theresults of existing studies and/or builds data from another source (e.g.patients logging their own experiences into such a website). The patientcould select their threshold of adverse outcome: (i) mild discomfort,(ii) severe discomfort, (iii) hospitalization, and (iv) death. Thewebsite would then display a 4×4 grid calibrated to display frequency ofadverse outcome. The upper left quadrant would be a general populationor placebo statistic since adverse outcomes happen even among healthypatients in the absence of medical treatment. The upper right coulddisplay frequency of adverse outcomes for patients takinganti-cholesterol medication but not pain medication. The lower leftcould display adverse frequency of adverse outcomes for patients takingpain medication but not anti-cholesterol medication. The lower rightwould display adverse outcomes for patients taking both medications withthis joint probability outcome reflecting a “thresholded” GerberStatistic. This could be color coded to make for especially clearinterpretation. After reviewing the graphical data presentation, thepatient may decide that the interaction risk is sufficiently low to beworth tolerating, that the drugs should be applied in sequence to avoida negative incremental impact of drug one on drug two (or vice-versa) orthat the joint probability of adverse outcome is too high to betolerable. Further, by calculating the Gerber Statistic for differentcombinations of drugs at different thresholds of adverse outcome, thepatient may make a more informed decision.

Referring now to FIG. 1 , an example architecture of a system 100 isshown. At least one user (e.g., a manager, a portfolio manager, trader,or analyst) can communicate with backend operations 170, including aserver 120, over a network using a computer 110, such as a personalcomputer, desktop computer, laptop computer, personal data assistant(PDA), mobile device (e.g., a cellular phone), tablet computer,telephone, smart phone, or any other computing device. The network canbe a local area network, wide area network, WI-FI network, or any othertype of connection between the server 120 and the computer 110. Althoughthe computer 110 is described as being used by the portfolio manager, itis intended that the label of a portfolio manager is not limited to anentity that has a supervisory role, but rather can include any entity,such as a trader, analyst, or investor, and each entity can have its owncomputer 110 for interaction with the system 100. The embodimentsdescribed herein use the terms investor, trader, manager, portfoliomanager, analyst, and user, though it is intended that these functionsand roles can be performed by or on behalf of any entity that instructs,uses, or implements the methods and systems described herein. In theexample embodiment, the portfolio manager can identify and propose newinvestments for purposes of portfolio construction.

The system 100 can also include an entry system 160, which can be acomponent of the server 120 or a separate, communicatively-coupleddevice, shown in the example configuration in FIG. 2 as a separateserver. The entry system 160 can allow the portfolio manager usingcomputer 110 to submit input data regarding variables as well as inputscontrolling backend operations 170. The entry system 160 can also beconfigured to automatically process input data regarding variables frominput data 150. The entry system 160 can also communicate with theserver 120 and any other components of the system 100.

At least one manager, such as a chief risk officer or a chief investmentmanager, can communicate with the server 120 over a network using acomputer 140, such as a personal computer, desktop computer, laptopcomputer, personal data assistant (PDA), mobile device (e.g., a cellularphone), tablet computer, telephone, smart phone, or any other computingdevice. The network can be a local area network, wide area network,WI-FI network, or any other type of connection between the server 120and the computer 140. In the example embodiment, the manager can monitorasset allocation and evaluate risk of an investment strategy. Themanager may reduce an allocation or impose a different portfolioconstruction based on an evaluation of diversification and risk.

The server 120 can transmit and receive information from the portfoliomanager's computer 110 and the manager's computer 140, and can receiveinput data 150 from additional sources. Input data 150 can include anydata about variables for purposes of measurement and analysis, and otherrelated information. The input data 150 can be imported directly intothe server 120, entry system 160 can transmit the input data 150 to theserver 120, or computer 110 and computer 140 can transmit the input data150 to the server 120. In some embodiments, the input data 150 caninclude real-time updates on stock prices, trade data from a data feed,historical data regarding one or more financial markets, dealer quotes,valuation services, models, good faith estimates or data from otherfinancial data monitoring services.

The server 120 can store information in a database 130. The database 130can be connected to the server 120 using a network, or alternatively,the server 120 and the database 130 can be integrated as a singlecomputing device. It is also understood that the server 120 and thedatabase 130 can each comprise multiple devices. The database 130 canmanage (e.g., store, maintain, delete, search, and retrieve) recordsregarding variables, analysis regarding the variables, and other relatedinformation. In some embodiments, the database 130 can also includerecords regarding portfolio construction or asset allocation. Thedatabase 130 can store time-series data including, but not limited to,data points regarding variables and other external data. The time-seriesdata in the database 130 can be for both current and historical data.

In the example embodiment, a system can compare two or more ideas,concepts, projects, or strategies, which may be implemented into thesystem as variables. Examples of these ideas, concepts, projects, orstrategies can include sport statistics, behavioral statistics,employment statistics, real estate statistics, deal codes, investmentstrategies, and/or any other measurable objective data. In anembodiment, the systems and methods can be used to assess therelationship between financial indicators (e.g., asset classes, sectors,or markets) implemented into the system as variables. A financialindicator implemented into the system as a variable can be based on, butis not limited to, an asset class, sector, index, market, geographicarea, note, corporate bond, municipal bond, stock, treasury stock,debenture, mutual funds, certificate of interest, certificate ofdeposit, derivative, commodity, currency, trust, put, call, straddle,option, investment in a partnership, investment in a limited liabilitycorporation, fixed income security, equity or debt security, any othertype of security or investment or any combination thereof.

Variable records may be stored in the database 130. Each record storedin the database 130 can include data points regarding the variable. Thedatabase 130 can store additional information in the record orassociated with the record. The additional information can include, butis not limited to, variable type, present variable value, and comments.The database 130 can store variable data points collected during thehistory of a particular variable, so that a user, such as the portfoliomanager or the manager, can query the database 130 to determine, insubstantially real-time, the behavior of a variable since it was firstentered into the system.

The portfolio manager via computer 110 and the manager via computer 140can communicate with the server 120 to add, modify, delete, transfer,associate, and update variable records in the database 130. Input data150 imported into the server 120 can also be used to update or otherwisemodify the variable records in the database 130. The portfolio managervia computer 110 or the manager via computer 140 can search the database130 for substantially real-time variable data points or for historicaldata. Additionally, the data can be aggregated based on any of theavailable fields for all date ranges. For example, the database 130 canaggregate all variable records based upon a particular criteria (e.g.,all variable records relating to an asset class can be aggregated).

An example process for measuring variables and the relationships betweenthem can be characterized according to (1) a filtering stage, (2) anevaluation stage, and (3) a monitoring stage. It is intended that thesestages are merely illustrative. The method is not limited to the orderof steps or stages described, and steps or stages may be omitted in someembodiments.

Each of the stages of the system 100 can be implemented by a softwaremodule executed by a processor via one or more of the computer 110,server 120, computer 140, or a combination thereof. The first stage canbe implemented in a filtering and collecting software module, the secondstage can be implemented in an evaluative and performance statisticssoftware module, and the third stage can be implemented in a monitoringsoftware module. These modules can function together with the database130 to provide data storage, evaluation, and monitoring of variables.

The storage of variable records in the database 130 allows for comparingmultiple variables with each other. In this embodiment, calculation of aGerber statistic may be performed, though it is understood that othertypes of statistical analysis may be performed in combination withcalculation of a Gerber statistic.

The system 100 can present information for display on computer 110 forthe portfolio manager or computer 140 for the manager regarding datapoints associated with a variable record in the database 130. Theportfolio manager or the manager can query the system 100 to analyze theGerber relationship between two or more variables, and the system 100can output this information for each variable.

The systems and methods described herein are related to those describedin the U.S. patent application Ser. Nos. 13/601,310 and 14/015,257,which are incorporated by reference in their entirety. For example, theuse of deal code records to monitor investments as taught in the '310and '257 Applications is another implementation of the current frameworkfor measuring relationships between variables. In the context of thesystems and methods of the '310 and '257 Application, each deal coderecord can be considered a variable and the investment monitoring systemcan measures the relationships between those variables.

Based on the Gerber relationships between variables, the system candisplay on a user interface the extent of a relationship between two ormore variables, as depicted in FIGS. 8-9 . The relationship can bedepicted in a format whereby variables moving in the same direction andhaving a positive relation may be depicted differently (e.g., differentsize, color, or shape) than those variables moving in a differentdirection and having a negative relation. In one example, the userinterface can present a treemapping of Gerber statistic values, wherebythe size of a nested rectangle can be indicative of the extent of aGerber relationship between two variables, whereby variables representedby larger rectangles are more related to other variables than thosevariables represented by smaller rectangles. In one alternative, thetreemapping of variable relationships can include only those variableshaving a positive relation or can perform filtering based on othercriteria. In another example, the analysis can be presented in acartographic generalization, whereby a geographic map is generated basedon the relationships and elevation can represent an extent of arelationship. In yet another example, the analysis can be presented in amulti-layer Venn diagram, whereby overlapping sections can represent theextent of a relationship between variables. In another example,different assets can be displayed in a spanning tree in accordance withtheir corresponding Gerber relationships. It is intended that anyrepresentation can be displayed, including the use of pictures, symbols,colors, and words, to show an extent of relationship between variables.

In some embodiments, the Gerber relationship between variables (e.g.,investments, assets classes, sectors, and markets) can be used toevaluate the co-movement of the variables. A diversity score can becalculated that represents an extent of co-movement between two or morevariables. For example, points can be allocated to represent thedirection and extent of a Gerber relationship between two or morevariables to generate a diversity score. Each variable can be allocatedwith a point for each instance where the variable has a negative Gerberstatistic (i.e., generally moves in opposite directions) with respect toanother variable. Variables can also be allocated with fractional pointsfor those negative relations that occur less than a hundred percent ofthe time (e.g., for a relation of −20%, a 0.2 can be awarded). Likewise,a negative point or fraction thereof can be applied each time a variablehas a positive relation (i.e., generally moves in the same direction)with respect to another variable. The total points for a variable can beconsidered a diversity score. In some configurations and embodiments ahigher diversity score is more favorable for some variables (e.g.,investments).

The Gerber relationship can be used in the context of portfolioconstruction. In constructing a portfolio, an investor determines how toallocate capital between various assets (e.g., equities, fixed incomesecurities, cash, real estate, currency, alternatives, commodities,collectibles, and derivatives) based upon risk tolerance or minimum rateof return. A portfolio with a high diversification of assets can subjectthe investor to lower risk for the same level of expected return, andthe Gerber relationship can be used to measure the diversification of aportfolio.

An established method for portfolio construction according tomean-variance optimization involves analyzing the risk of potentialinvestments using expected return, expected variance, and expectedcovariance. This method is described in further detail in “PortfolioSelection” and “Portfolio Selection: Efficient Diversification ofInvestments,” incorporated herein by reference in their entirety. Theportfolio can then be optimized based on risk tolerance or returnrequirements. When applying this method, the Gerber relationship can beused in place of correlation to provide a more accurate measure ofexpected covariance than the conventional measure of expected covarianceand/or expected semi-variance.

In allocating capital among various assets with different levels ofrisk, an investor might focus on achieving the best possible rate ofreturn for the portfolio without exceeding a risk limit, which isaffected by the diversification of the individual assets in theportfolio. As described herein, risk can be described as an estimatedprobability of a return below a negative threshold. In other words, theinvestor typically desires the best possible return for a given risklevel. In some scenarios, an investor may seek the minimum amount ofrisk based on a given return target. The risk of a portfolio's return isrelated to the variance of its return, and so a goal of portfolioconstruction is to create a portfolio with a high return and a minimizedvariance. But the variance of a portfolio also depends on thecovariances between the individual investments. Accordingly, optimalportfolio construction accounts for the co-movement of investments.

Conventional portfolio construction methods attempt to determine arisk-adjusted return of a portfolio of investments using eachinvestment's expected return and covariance with the other investmentsin the portfolio. Traditionally, covariance of two investments is basedon correlation and may be calculated as follows:

Cov(R _(X) ,R _(Y))=σ_(X)σ_(Y)ρ_(XY)

Where RX denotes a return of the first investment, RY denotes a returnof the second investment, σX denotes a standard deviation of the returnof the first investment, σY denotes a standard deviation of the returnof the second investment, and ρXY denotes a correlation value betweenthe first and second investments. A correlation value must always be anumber between −1 and 1, whereby a correlation of 1 indicates that theinvestments move perfectly together, a correlation of 0 indicates thatthe investments move independently from each other, and a correlation of−1 indicates that the investments move perfectly in opposite directions.Conventional methods use this covariance formula to calculate thestandard deviation of the returns from a multi-investment portfolio,whereby the standard deviation may represent an indicator of risk forthe portfolio.

The system can use Gerber relationships to calculate covariance ofinvestments instead of the conventional methods that rely oncorrelation. In some embodiments, the system can use Gerberrelationships to calculate a covariance matrix comparing each possiblepair of investments in a portfolio. A Gerber statistic can provide aco-movement measure in the same units and range as a conventionalcorrelation calculation (e.g., a number between −1 and 1). As a result,the Gerber statistic can easily replace the conventional correlationmeasurement when calculating the covariance of a portfolio. Using theGerber statistic as a replacement for correlation, the same expectedvariances may be used to calculate covariances or semi-variances, whichcan then be used with the same expected returns to identify amean-variance optimal allocation for each investment in the portfolio.The resulting portfolio construction or optimization will produceimproved results because of the previously discussed advantages thatmeasuring the Gerber statistic has over conventional correlation.

An investor can use a computer system, such as system 100, to calculatethe expected return of a proposed portfolio or an existing portfolio.The investor can input the portfolio's investments into the system,which can access historical data about the investments and calculate thenecessary Gerber statistic. The computer system can assess the Gerberstatistic and display a figure, number, scale, or other graphic to theinvestor about the risk in the investments. Based upon an input of acapital amount to invest, the systems can determine how to allocate thecapital based upon the investor's acceptable level of risk or targetreturns. For example, when attempting to maximize returns of a portfoliofor a given level of risk, the computer system can vary the weightingsof different investments to find the best possible expected returnswithout exceeding the given level of risk. The system may then allocatecapital based upon the weighting of those investments to maintain theappropriate risk-reward levels. In one embodiment, the investor canadjust a level of acceptable risk, and the computer system can suggest anew weighting of the investments to maximize returns for that risklevel. Upon a confirmation by the investor, the system can automaticallyallocate the capital accordingly.

As described herein, risk may refer to an estimated probability of areturn below a negative threshold. Furthermore, different end users mayhave different risk tolerances and/or risk preferences. For instance, along-term investor may view a −20% return worse than a +20% returnbecause the latter takes greater returns to recover from. Therefore, themethods and systems described herein can be used for investmentstrategies with stop losses, managing assets where their “downsidevolatility” is believed to be more costly than “upside volatility,”hedging of fixed strike exotic options, issuance of structured productswith capital floors where the hedger takes residual gap risk, orpotential extensions into risk allocation and portfolio sizing usingother protocols, such as Kelly Criterion.

The Gerber statistic is a robust measure of correlation between datapoints representing different assets. The Gerber statistic allows aprocessor to analyze (e.g., count) the proportion of simultaneousco-movements in series of data points when their amplitudes exceeddata-dependent thresholds. The Gerber statistic described herein isunlike conventional methods, such as the Kendall's Tau or the standardPearson correlation that are sensitive to outliers or the Spearmancorrelation that relies on ranking observations.

As will be described herein, the one or more versions of the Gerberstatistic are neither affected by extremely large or extremely smallmovements. Therefore, the Gerber statistic is suited to analyzefinancial time series data since these time series data can be noisy,include fluctuations, and/or exhibit extreme movements (e.g., suddenspikes or asset price re-basing on material incremental information). Acomputer server, such as the computer system 100 depicted in FIG. 2 ,can utilize the Gerber statistic to calculate an estimate of acovariance matrix that is suitable for portfolio optimization.

Portfolio construction and optimization, such as the Markowitz methoddescribed herein, relies heavily on the availability of the matrix ofcovariances between securities' returns. In some configurations, thehistoric covariance matrix is used as an estimate for future covariancematrix. Various models have been used to ease the computational burdenand to improve statistical properties of covariance matrix estimates.However, many conventional methods suffer from a technical shortcomingwhen estimating covariance matrices. For instance, conventional methodsuse product-moment-based estimates that are inherently inefficient ifthe underlying distribution is prone to containing extreme measurementsor outliers.

These shortcomings cause incorrect results or require heavycomputational resources when applied to financial data. For instance,financial time series data are particularly noisy, and a computeranalyzing the financial time series data using conventional methods caneasily misinterpret the noise as information. One consequence, forexample, is that the correlation matrix estimates (even ones constructedusing robust techniques) often have non-zero entries corresponding toseries that in fact have no meaningful correlation. The correlationestimates can also be distorted if the series contains extremely large(positive or negative) observations.

The Gerber statistic versions described herein provide a robust methodfor computing a co-movement measure that ignore fluctuations below acertain threshold, while simultaneously limiting the effects of extrememovements. For instance, r_(tk) may represent the return of security kat time t (e.g., for k=1, . . . , K securities and t=1, . . . , T timeperiods). For every pair (i,j) of assets for each time t, the Gerberstatistic may convert each return observation pair (r_(ti), r_(tj)) to ajoint observation m_(ij)(t) defined using the equation depicted below:

${m_{ij}(t)} = \left\{ \begin{matrix}{{{{+ 1}{\ }{if}\ r_{ti}} \geq {{+ H_{i}}\ {and}r_{tj}} \geq {+ H_{j}}},} \\{{{{+ 1}\ {if}\ r_{ti}} \leq {{- H_{i}}\ {and}r_{tj}} \leq {- H_{j}}},} \\{{{{- 1}{if}r_{ti}} \geq {{+ H_{i}}\ {and}{\ }r_{tj}} \leq {- H_{j}}},} \\{{{{- 1}\ {if}\ r_{ti}} \leq {{- H_{i}}\ {and}\ r_{tj}} \geq {+ H_{j}}},} \\{{0{otherwise}},}\end{matrix} \right.$

In the depicted equation, H_(k) represents a threshold for security k.The joint observation m_(ij)(t) is therefore set to +1 if the series iand j simultaneously satisfy their thresholds in the same direction attime t; to −1 if they satisfy their thresholds in opposite directions attime t, or to zero if at least one of the series does not satisfy itsthreshold at time t.

A pair for which both components satisfy their thresholds while movingin the same direction can also be referred to as a concordant pair(e.g., co-movement), and one whose components satisfy their thresholdswhile moving in opposite directions can be referred to as a discordantpair.

In a configuration, the system utilizing the Gerber statistic may setthe threshold H_(k) for security k to be:

H _(k) =Cσ _(k)

Where c is some fraction (e.g., ½) and σ_(k) is the sample standarddeviation of the return of security k. The system may also consider awindow of time over which the standard of deviation is calculated (e.g.,a period for each individual return). For instance, the standarddeviation value for an asset calculated for 1 day of minute-by-minutevalue changes in USDJPY may differ from the standard deviation of thesame asset for 10 years of monthly returns. In alternativeconfigurations, more robust measures than standard deviation can be usedfor the threshold computation. The Gerber statistic for a pair of assetscan then be defined as:

$\begin{matrix}{g_{ij} = \frac{{\sum}_{t = 1}^{T}{m_{ij}(t)}}{{\sum}_{t = 1}^{T}{❘{m_{ij}(t)}❘}}} & (1)\end{matrix}$

Letting n^(c) _(ij) be the number of concordant pairs for series i andj, and letting n^(d) _(ij) be the number of discordant pairs, it can beshown that Equation (1) is equivalent to:

$g_{ij} = {\frac{n_{ij}^{c} - n_{ij}^{d}}{n_{ij}^{c} + n_{ij}^{d}}.}$

Since this statistic relies on counts of the number of simultaneoussatisfaction of thresholds (and not on the extent to which thethresholds are satisfied), it may be less sensitive to extreme movementsthat distort product-moment-based measures. At the same time, since aseries must exceed its threshold before it becomes a candidate to becounted, the measure is also less sensitive to small movements that maysimply be noise.

To generate the desired matrix, the system may define R∈R^(T×K) as thereturn matrix having r_(th) in its t^(th) row and k^(th) column. Thesystem may also define U as a matrix with the same size as R havingentries u_(tj) such that:

$u_{tj} = \left\{ \begin{matrix}1 & {{{{if}r_{tj}} \geq {+ H_{j}}},} \\0 & {{otherwise}.}\end{matrix} \right.$

With these definitions, the matrix of the number of samples that exceedthe upper threshold will become N^(u)u=U^(t)U. In this example, the ijelement n^(uu) _(ij) of N^(UU) is the number of samples for which bothtime series i exceeds the upper threshold and for which time series jsimultaneously exceeds the upper threshold.

Similarly, the system may define D as the matrix with the same size as Rhaving entries d_(tj) such that:

$d_{tj} = \left\{ \begin{matrix}1 & {{{{if}r_{tj}} \leq {- H_{j}}},} \\0 & {{otherwise}.}\end{matrix} \right.$

With this definition, the matrix of the number of samples that are underthe lower threshold will become N^(DD)=D^(t)D. As can be inferred, thismethod may utilize the useful property that ij element n^(DD) _(ij) ofN^(DD) is the number of samples for which both time series i is belowthe lower threshold and for which time series j is simultaneously belowthe lower threshold. Accordingly, the matrix containing the number ofconcordant pairs becomes:

N _(CONC) =N ^(UU) +N ^(DD) =U ^(T) U+D ^(T) D

Furthermore, the matrix containing the numbers of discordant pairsbecomes:

N_(DISC) =U ^(T) D+D ^(T) U

The system may then generate the Gerber matrix “G” (e.g., the matrixthat contains g_(ij) in its i^(th) row and j^(th) column) in theequivalent matrix form:

G=(N _(CONC) −N _(DISC))

(N_(CONC)+N_(DISC))

Where the symbol Ørepresents the Hadamard (elementwise) division. Tosimplify the description of various versions of the Gerber statistic, itis useful to consider the following graphical representation for therelationship between two securities:

UD UN UU ND NN NU DD DN DU

As depicted above, the rows represent categorizations of security i. Thecolumns represent categorizations of security j. The boundaries betweenthe rows and the columns represent the chosen thresholds. The letter Urepresents the case in which a security's return lies above the upperthreshold (e.g., is up). The letter N represents the case in which asecurity's return lies between the upper and lower thresholds (e.g., isneutral). The letter D represents the case in which a security's returnlies below the lower threshold (e.g., is down). In a non-limitingexample, if at time t, the return of security i is above the upperthreshold, this observation lies in the top row. If, at the same time t,the return of security j lies between the two thresholds, thisobservation lies in the middle column. Therefore, this observation liesin the UN region.

When executed iteratively and over a period of time (e.g., t=1, . . . ,T), there will be observations scattered over the nine regions. Letn^(pq) _(ij) be the number of observations for which the returns ofsecurities i and j lie in regions p and q. Respectively, for p, q€{U, N,D}. With this notation, the system can obtain another equivalentexpression for the Gerber statistic as:

$g_{ij} = {\frac{n_{ij}^{UU} + n_{ij}^{DD} - n_{ij}^{UD} - n_{ij}^{DU}}{n_{ij}^{UU} + n_{ij}^{DD} + n_{ij}^{UD} + n_{ij}^{DU}}.}$

The correlation matrix constructed from the Gerber statistic describedin the patent applications to which this application claims priority andas defined in Equation (1) may sometimes lead to results that are notpositive semidefinite (PSD). If the system encounters a covariancematrix that is not PSD, then the system may construct a portfolioindicating a negative risk. As a result, the system may indicate anarbitrarily large position based on the mistaken belief that risktolerances will not be breached, which may lead to erroneous results.

As a result, the system may also utilize a few alternative methods. In afirst non-limiting example, the system may use:

$g_{ij}^{(1)} = {\frac{{\sum}_{t = 1}^{T}{m_{ij}(t)}}{T - n_{ij}^{NN}}.}$

This can be written in terms of the alternative notation as:

$\begin{matrix}{g_{ij}^{(1)} = {\frac{n_{ij}^{UU} + n_{ij}^{DD} - n_{ij}^{UD} - n_{ij}^{DU}}{T - n_{ij}^{NN}}.}} & (2)\end{matrix}$

The above equation (Equation (2)) is also referred to herein as GerberStatistic (GS1), which is a different version of the Gerber statistic(GS). Another version, Gerber Statistic 2 (GS2), can be defined as:

$\begin{matrix}{g_{ij}^{(2)} = \frac{n_{ij}^{UU} + n_{ij}^{DD} - n_{ij}^{UD} - n_{ij}^{DU}}{\sqrt{n_{ij}^{(A)}n_{ij}^{(B)}}}} & (3)\end{matrix}$

-   -   where the n^((A)) _(ij) and n^((B)) _(ij) in the denominator are        defined as:

n _(ij) ^((A)) =n _(ij) ^(UU) +n _(ij) ^(UN) +n _(ij) ^(UD) +n _(ij)^(DU) +n _(ij) ^(DN) +n _(ij) ^(DD),

n _(ij) ^((B)) =n _(ij) ^(UU) +n _(ij) ^(NU) +n _(ij) ^(DU) +n _(ij)^(UD) +n _(ij) ^(ND) +n _(ij) ^(DD).

Let Q=N_(CONC)−N_(DISC); and let q=the √{square root over (Diag(Q))} tobe the vector of square roots of the diagonal element of Q (which areall positive). Therefore, it can be shown that GS2 can be written in thematrix form:

G ⁽²⁾=(N _(CONC) −N _(DISC))

(qq ^(T))

Written differently (letting J=J^(t)) be the diagonal matrix with theinverse of the i^(th) element of q in its i^(th) diagonal position wouldlead to:

G ⁽²⁾ =J ^(T)(N _(CONC) −N _(DISC))J.

Portfolio optimizers may require the covariance matrix of securities'returns to be positive semidefinite. The methods and systems describedherein (e.g., Gerber matrix) can be used as a robust version of thecorrelation matrix from which a corresponding robust version of thecovariance matrix can be constructed. The system may use this version ofthe covariance matrix in a portfolio optimizer. Therefore, the systemmay require the Gerber matrix to be positive semidefinite.

The Gerber matrix can be viewed as a matrix ratio whose numerator matrixis Q=N_(CONC)N_(DISC) and whose denominator matrix depends on theparticular alternative chosen. If the numerator matrix is positivesemidefinite, the Gerber matrix will be positive semidefinite if thedenominator is positive semidefinite. Therefore, to establish that thegiven alternatives are positive semidefinite the following proves thatthe numerator matrix is positive semidefinite.

From the definitions of N_(CONC) and N_(DISC), the numerator matrix canbe written in the following squared form:

$\begin{matrix}Q & {= {N_{CONC} - N_{DISC}}} \\ & {= {{U^{\top}U} + {D^{\top}D} - {U^{\top}D} - {D^{\top}U}}} \\ & {= {\left( {U - D} \right)^{\top}\left( {U - D} \right)}}\end{matrix}$

Therefore, for arbitrary but non-zero X:

x ^(T) Qx=x ^(T)(U−D)^(T)(U−D)x=u ^(T) u≥0

As a result, the numerator matrix will be positive semidefinite. Forcertain cases, it is possible to extend this analysis to show that theGerber matrix itself is positive semidefinite. For example, in thesecond alternative form:

$\begin{matrix}{x^{\top}G^{(2)}x} & {= {x^{\top}J^{\top}HJx}} \\ & {= \ {{x^{\top}{J^{\top}\left( {U - D} \right)}^{\top}\left( {U - D} \right)Jx} = {{u^{\top}u} \geq {0.}}}}\end{matrix}$

GS1 also produces positive semidefinite correlation matrices. This canbe proven by noting that the numerator matrix Q is positive semidefiniteas shown above, and the Hadamard denominator matrix is a positive matrixitself.

The system may also use an optimal shrinkage estimator protocol. Thesystem may use the methods described herein to calculate covariancebetween a pair of assets. For instance, in a non-limiting example of asample covariance matrix method described below, let r_(j,t) denote thehistorical return for asset i at time period t and the average returnover the time ranging from t=1 to t=T to be ri:

${\overset{¯}{r}}_{i} = {\frac{1}{T}{\sum\limits_{t = 1}^{T}{r_{i,t}.}}}$

Then the sample covariance between a pair of assets can be estimatedvia:

${Co{v\left( {r_{i},r_{j}} \right)}} = {{\frac{1}{T - 1}{\sum\limits_{t = 1}^{T}{\left( {r_{i,t} - {\overset{¯}{r}}_{i}} \right)\left( {r_{j,t} - {\overset{¯}{r}}_{i}} \right)}}}\overset{def}{=}{{\overset{\hat{}}{\sigma}}_{ij}.}}$

The historical covariance matrix for N assets can be specified viaevaluating the above equation for pairs of i,j assets or:

${\hat{\sum}}_{HC}{= {\begin{bmatrix}{\overset{\hat{}}{\sigma}}_{11} & {\overset{\hat{}}{\sigma}}_{12} & \cdots & {\overset{\hat{}}{\sigma}}_{1N} \\{\overset{\hat{}}{\sigma}}_{211} & {\overset{\hat{}}{\sigma}}_{22} & \cdots & {\overset{\hat{}}{\sigma}}_{2N} \\ \vdots & \vdots & \ddots & \vdots \\{\overset{\hat{}}{\sigma}}_{N1} & {\overset{\hat{}}{\sigma}}_{N2} & \cdots & {\overset{\hat{}}{\sigma}}_{NN}\end{bmatrix}.}}$

The estimated covariance matrices can then be obtained from thehistorical correlation matrix:

{circumflex over (Σ)}_(HC)=diag({circumflex over (σ)})Ĉ_(HC)diag({circumflex over (σ)})

Where σ is an N×1 vector of sample standard deviation of the historicalasset returns, expected future returns, or expected future returns aspriced by various derivative markets and CHC is the sample correlationmatric of the historical asset returns. In another non-limiting example,such as the single-index method described below, the system may use aSharpe's single-index model. The single-index model assumes the returnof the an individual stock i is related to the return of a stock marketindex m, as follows:

r_(i)=α_(i)+β_(i) r _(m)+∈_(i)

Where α_(i) is the excess return that is independent of the marketchanges, β_(i) is a measurement of the sensitivity of asset i's returnto the market index return, and ∈_(i) is the residual term with

[∈i]=0. The single-index covariance estimator assumes that the residualterms between assets are independent e.g., ∈i and ∈_(j) are independentfor i and j pairs such that:

[∈_(i)∈_(j)]=0,∀i,j(i≠j).

Given this assumption, one can show the variance of an asset i is:

${\sigma_{i}^{2} = {\begin{matrix}\underset{︸}{\beta_{i}^{2}\sigma_{m}^{2}} \\{{Systematic}{risk}}\end{matrix} + \begin{matrix}\underset{︸}{\sigma_{\varepsilon_{i}}^{2}} \\{{Asset}{specific}{risk}}\end{matrix}}},$

Where σ_(m) ² is the variance of the market returns and σ_(ei) ² is thevariance of ∈_(i). The covariance between two assets i and j is givenby:

σ_(ij)=β_(i)β_(j)σ_(m) ² ∀i,k,i≠j,

-   -   and the estimated covariance matrix implied by such model is:

{circumflex over (Σ)}_(SI)={circumflex over (β)}{circumflex over(β)}^(T)σ_(m) ²+diag({circumflex over (σ)}_(∈) ²),

Where β={β₁ . . . β_(N)}^(T) denotes a vector of estimated betas and thefollowing represent a vector of estimated variances of residual termsfor each asset:

{circumflex over (σ)}_(∈) ²={{circumflex over (σ)}_(∈1) ², . . . ,{circumflex over (σ)}_(∈N) ²}

In some configurations, the system may utilize a shrinkage method thatachieves a balance between the sample covariance and single-indexmethods described herein. For instance, the system may use a shrinkageparameter of:

a∈{1,0}

This shrinkage parameter may balance between the two approachesdiscussed herein, as depicted below:

Σ_(SM) =aΣ _(SI)+(−a)Σ_(HC).

The system may find the optimal shrinkage parameter a via minimizing theFrobenius norm between the asymptomatically true covariance matrix andshrinkage estimate as depicted below:

$a^{*} = {\arg\underset{a}{\min}{{a{\sum_{SI}{{+ \left( {1 - a} \right)}{\sum_{HC}{- \sum}}}}}}_{F}^{2}}$

Referring now to FIGS. 5A and B, different versions of the Gerberstatistic (GS, GS1 and GS2) are depicted by equations 510-530. Theseequations correspond to analyzing data represented by the data pointsdepicted within the graph 500 (FIG. 5B). In FIG. 5A, the Gerberstatistic (GS) is represented by the equation 510, which indicates whichdata points depicted in FIG. 5B are used to calculate the Gerberstatistic. GS1 is represented by equation 520, which indicates whichdata points depicted in FIG. 5B are used by GS1. GS2 is represented byequation 530, which indicates which data points depicted in FIG. 5B areused by GS2.

In the embodiment depicted in FIGS. 5A and B, different versions of theGerber statistic are used to analyze data associated with two assets (Aand B). Each data point within the graph 500 (depicted in FIG. 5B) mayrepresent a transformed (e.g., discretized or normalized) valueassociated with each asset. For instance, data points reflecting stockprices for different times may be transformed into a range of −1 to 1.These data points are shown in the graph 500 and arranged based on theirvalues with respect to the axis 540 and 550. As depicted, the equation510 accounts for a difference between the data points within the boxes506 and 504 compared with data points within the boxes 502 and 508. Inthe denominator, the equation 510 accounts for a difference between thetotal number of data points and the data points outside the boxes 502,504, 506, and 508 (e.g., data points within the box 509).

A difference between GS and GS1 (represented by the equation 520) isthat while the numerator of both equations are the same, the denominatorof the equation 520 accounts for more data points. Specifically, theequation 520 accounts for all the data points excluding the data pointswithin the box 509 (where both asset A and B are below the threshold).Effectively, the equation 520 also accounts for data points 560 a-f,which are not considered in the equation 510. This modification allowsfor the system to account for more data points while maintaining PSDresults.

GS2 (represented by the equation 530) shares the same numerator as theother equations. The equation 530 includes the square roots of the datapoints where asset A satisfies the threshold (e.g., every time that adata point for asset A is above the threshold on either sides, whetheris it negative or positive) multiplied by the square root of all datapoints where asset B satisfies a threshold (e.g., every time that assetB is above the threshold on either sides, whether is it negative orpositive). This modification allows for the system to account for moredata points while maintaining PSD results.

Some aspects of the present disclosure discuss a 2×2 matrix to beanalyzed (e.g., a matrix that analyzed data points and determineswhether they are above or below a threshold). However, it is understoodthat the methods and systems described herein can apply to other numberof variables too. For instance, the methods and systems described hereincan use a 3×3 matrix where each variable is bucketed into the followingthree categories: above-threshold (e.g., box 502), below-threshold(e.g., box 504), and between-threshold (box 509). Using this data aserver can identify many insights. For instance, data points that are inbetween thresholds may describe how infrequently the assets movesufficiently and may also indicate outcomes of smaller “drift” moves. Insome embodiments, a graphical user interface may display the data pointson the end-user's device, such as depicted in FIG. 9 where the datapoints of all three categories are displayed.

Classical portfolio construction optimization methods generally relyupon covariance matrix methods. Covariance of assets can bemathematically defined as the multiplication of the standard deviationof each asset by the correlation of the two asset returns. Using themethods discussed herein, the correlation, as used in the classicalportfolio construction optimization methods, can be replaced by theGerber relationship calculated using one or more versions of the Gerberstatistic because the Gerber statistic is more efficient and accuratemeasurement of co-movement between two assets. Therefore, an optimizercan change covariance with Gerber statistic (any of the versions) toachieve better results. Using the methods and systems described herein,performance of an asset can be monitored, such that investments can bemanaged while limiting the risk (e.g., preventing or minimizing theprobability of the return from going below a pre-determined threshold).Therefore, utilizing the Gerber statistic will improve classicalportfolio optimization methods by maximizing return and minimizingdrawdown.

Moreover, the methods discussed herein also provide a semi-variancerelationship between the analyzed data points. Semi-variance is definedas the expected squared deviation from a threshold, d, usually chosen aseither 0 or the mean value for time-series data to be analyzed, asdepicted in the following equation:

S=E{(r−d)_∧2}

Where E is the expectation operator, r is the portfolio return, and thenegative part function is:

$\left. x \right.\_ = \left\{ \begin{matrix}x & {{{if}x} < 0} \\0 & {{{if}x} \geq 0}\end{matrix} \right.$

In the covariance methods, the upside risk and the downside risk aremathematically defined the same. For instance, the risk of an assetincreasing or decreasing by 2% is weighted equally by covarianceoptimization methods. In contrast, a semi-variance method weights theserisk differently. For instance, a 2% chance of an asset increasing maybe deemed more important than a 2% chance of the same asset decreasing(or vice versa).

The semi-variance method discussed herein may yield better resultsbecause the return distribution for financial time-series data istypically not symmetrical and/or because investor preferences maydiffer. The Gerber statistic can be used in conjunction with optimizersto construct and optimize portfolios.

In general, the objective of the mean-variance portfolio problem is tochoose a portfolio x so as to:

minimize V=x ^(T) Cx

subject to μ^(T) x=E,

Ax=b,

x≥0,

for all E∈[E _(min) , E _(max)].

Using the methods described herein, the system may minimizesemi-variance (or alternatively, the variance below a predeterminedvalue). Therefore, instead of minimizing V in the problem above, thesystem may minimize S in the below equation:

S=

[(r _(p) −−d)²_]

-   -   where rp is the portfolio return, d is a downside threshold, the        symbol E represents the expectation operator, and the negative        sign in the subscript denotes the absolute value of the negative        part, that is:

$\left. x \right.\_ = \left\{ \begin{matrix}{❘x❘} & {{{{if}x} < 0},} \\0 & {{{if}x} \geq 0.}\end{matrix} \right.$

To perform downside optimization of S, in the above equation (1), thesystem may use:

$S = {\frac{1}{T}{\sum\limits_{t = 1}^{T}\left( {{r_{p}(t)} - d} \right)_{\_}^{2}}}$

-   -   where r_(p)(t) is the return of the portfolio at time t.

When using a semi-variance method, the system may use the portfolio'sexpected return in place of d (in equation 1). Therefore, the downsideoptimization is replaced by a semi-variance optimization. Specifically,the system may define R as the T x n matrix of historical securityreturns:

$R = \begin{bmatrix}r_{11} & r_{12} & \cdots & r_{1n} \\r_{21} & r_{22} & \cdots & r_{2n} \\ \vdots & \vdots & \ddots & \vdots \\r_{T1} & r_{T2} & \cdots & r_{Tn}\end{bmatrix}$

That is, the element in row t and column j is the return of the j^(th)security in period t. The securities' returns in excess of their meanscan be defined as:

R−ιμ^(T),

-   -   where μ is the n-vector of mean returns and ι is an        appropriately dimensioned vector of ones. Therefore, the        time-series of portfolio returns below the mean is:

[(R−ιμ^(T))x]_,

Accordingly, the portfolio's semi-variance can be written as:

$\begin{matrix}{S = {{{\frac{1}{T}\left\lbrack {\left( {R - {\iota\mu^{\top}}} \right)x} \right\rbrack}_{–}^{\top}\left\lbrack {\left( {R - {\iota\mu^{\top}}} \right)x} \right\rbrack}\text{\_.}}} & (2)\end{matrix}$

-   -   whereby defining the matrix as:

$B = {\frac{1}{\sqrt{T}}\left\lbrack {R - {\iota\mu^{\top}}} \right\rbrack}$

-   -   and the two variables y and z are defined as y−Bx and:

z=y_.

With these definitions, the semi-variance defined in equation (2)becomes S=z^(t)z. The system can then reformulate the semi-varianceproblem as a minimization of a simple square subject to a new set ofconstraints, as shown below:

minimize S=z ^(T) z

subject to μ^(T) x=E,

AX=b,

Bx−y+z=0,

x,y,z≥0,

for all E∈[E _(min) , E _(max)],

The Gerber relationship can be calculated based on the coordinatedmovements of multiple variables using one or more versions of the Gerberstatistic described herein (e.g., GS, GS1, and/or GS2). Referring toFIG. 2 , this analysis can be performed by a data processing system(e.g., the system depicted in FIG. 1 ), in step 210. In one embodiment,an input filter may specify both the number of observation periods and athreshold value that the measured movement must exceed to be consideredas a qualifying event. For example, the analysis may consider theinstances over the last 25 days where both variables moved over apredefined amount (e.g., 1% of the value) in the same day. For each ofthe variable records in the analysis, the system 100 can then comparethe filtered results of each variable record to each of the othervariable records, in step 220. The results, which may indicate thefrequency of similar behavior, can be used to determine the uniqueness,with regard to performance, of each variable when compared to the otherindividual variables in the analysis. The analysis of step 230 can beperformed by computer 110, computer 140, server 120, or any combinationthereof. The results can be displayed, in step 230. For example,computer 110 or computer 140 can present the comparison of variables.

In some situations, groups of variables may exhibit related performanceover time. For example, a collection of variables associated with onecharacteristic (e.g., various investments associated with the same assetclass) may exhibit a pattern of performance when compared to acollection of variables associated with a second characteristic (e.g.,investments associated with a different asset class). Accordingly, thesystems and methods described herein can measure the Gerberrelationships between a first group of variables and a second group ofvariables.

The methods and systems described herein can be used to construct acustomized portfolio and dynamically reallocate assets to be invested inan automated fashion. The system may provide an electronic platform inwhich a robotic advisor (e.g., virtual advisor or robo-advisor) caningest investment preferences, goals, list of investment vehicles, andother relevant information from a user (e.g., portfolio manager,investor, or any other party interested in constructing a portfolio).The robo-advisor can construct a portfolio using the one or moreversions of the Gerber statistic discussed herein applied via one ormore optimization methods that utilize the user's inputted preferencesto maximize returns.

The methods and systems described herein can be used to create areturn-linked structured product. In a non-limiting example, an investormay have a defined capital pool of $100. Such an amount may prevent theinvestor from “shorting” or “leveraging” any asset without an exhaustiveand impractical credit analysis. As a result, the inventor may havethree choices: (i) a long-only portfolio with zero leverage (which maynot be optimal and may incentivize buying higher-leverage assets withinthe portfolio even if those assets offer inferior risk/reward); (ii) buyputs or calls to achieve defined-loss leverage and shorts (this strategywould introduce theta decay and added complexity which may not beoptimal); or (iii) buy a structured product for $100 where acounterparty executes a strategy on their behalf. Assuming that thestrategy could possibly go “negative” and incur losses beyond theinitial $100, the risk to the counterparty may be embedded in the priceof the product, which is undesirable to the investor.

Using the methods and systems described herein, the investor's assetscan be managed better in pricing catastrophic capital destruction casesthan other methods of portfolio construction. Specifically, using theGerber statistic discussed herein, counterparties (e.g., banks) couldprice the above-described risk more efficiently. As a result, theinvestor could get access to a broader range of investments/strategiesat a more reasonable price.

In a method of portfolio construction, as shown in FIG. 3 , a computersystem (e.g., system 100 shown in FIG. 1 ) can receive an input ofpotential investments from a user or another computer (e.g.,interconnected computers/servers automatically constructing aportfolio), in step 310. The input can include an identification ofdifferent asset classes, sectors, markets, investment strategies, orparticular investment vehicles. The system can also receive anacceptable level of risk, in step 320. The acceptable level of risk canbe determined by the user, or the system may use a default level. Basedupon the identified investments and the level of risk, the computersystem can calculate an expected return for the potential investmentsusing one or more versions of the Gerber statistic, in step 330. Thesystem may calculate various weightings of the investments to determinehow to allocate capital between these investments to achieve the maximumlevel of return while satisfying the acceptable rate of risk. The systemmay evaluate a series of scenarios in which different amount of capitalis allocated to different assets to identify which scenario yields thebest return. The system may use multiple different expected returnassumptions weighted by some probability of each expected return setbeing realized over the investment horizon. The system may then receivean input of an amount of capital, in step 340. The system can allocatethe capital to the inputted investments based upon the calculations, instep 350.

In a non-limiting example, a user accesses an electronic platform (e.g.,website) hosted or otherwise functionally controlled by the system. Theuser may use various input elements to enter a list of investments,assets, deal codes, investment strategies, and/or asset classes (e.g.,cash, stocks, and gold). The user may also indicate a risk tolerance(e.g., low, medium, or high risk indicating aggressive investing). Thesystem may use one or more versions of the Gerber statistic to constructa portfolio for the user. For example, the system may display anexplanation that because the user has chosen a conservative (low risk)investment strategy, the system has optimized a unique portfolio for theuser that includes 40% cash, 30% S&P investments, 20% gold, and 10%aggressive ETFs. The system may also indicate a percentage of capitalallocation for different S&P stocks. For instance, the system mayrecommend that the user allocates half of the capital to be allocatedthe S&P stocks (15% of the total investment) into a particular stock anddivide the other half into five different stocks.

The electronic platform displaying the recommendations may includeinteractive elements, such that the user can override/revise therecommendations. Upon detecting a change, the system may re-calculatethe projected/simulated return. In some configurations, the system may,upon receiving proper authorization from the user, allocate the user'scapital to the recommended investment vehicles by creating an accountfor the user. This method may be used for anyone who desires toconstruct a portfolio and maximize returns subject to a specific set ofconstraints (e.g., given a unit/preference of risk or predeterminedinvestments).

The methods and systems described herein can also be used to dynamicallyreallocate assets within a portfolio. In this way, the system mayoptimize passive investment vehicles for users. For instance, the systemmay use one or more versions of the Gerber statistic to calculate arelationship between assets within a portfolio. Using the calculatedrelationships, the system may automatically customize a portfolio inaccordance with various criteria. For instance, the system may analyzevarious assets (stocks) included within an exchange traded fund (ETF),structured product, and/or exchange traded product (ETP)and calculate aGerber relationship for each asset using the methods discussed herein.The system may then calculate an expected return within a definedtimeline for the ETF and determine whether the expected return satisfiesa threshold. The threshold may be inputted by a portfolio manager or asystem administrator. The threshold may indicate an expected returnvalue or may indicate a risk value associated with the ETF, ETP, and/orstructured product. When the system determines that the assets withinthe ETF do not satisfy the threshold, the system may dynamically revisethe assets within the ETF. For instance, the system may iterativelysimulate different allocations to different assets within the ETF.

The system may periodically monitor the ETF and dynamically revise itscontent in accordance with various rules and thresholds in order toadapt to predetermined themes (e.g., ETFs directed towards or isolatedfrom an industry or a sector) or adapt to ongoing market movements andtrends. In this way, investors can invest in a dynamic ETF where thesystem periodically revises the content of the ETF to maximize thereturn.

In an alternative embodiment, as shown in FIG. 4 , a computer system(e.g., system 100 shown in FIG. 2 ) can determine the risk of a proposedor existing portfolio based upon inputs. The system can receive an inputof investments, in step 410. For instance, a user (e.g., investor or aportfolio manager) can enter a list of desired investments (e.g., stocksand ETFs) and the system may display visual aid to describe the riskassociated with the portfolio (e.g., FIGS. 6-9 ).

The system can also receive an amount of capital for each investment, instep 420. For instance, the user can also enter an amount of capitalallocated (or desired to be allocated) to each investment.Alternatively, the user can provide a total amount of capital to beallocated to the investments.

The system can then calculate an expected return for the investmentsusing one or more versions of the Gerber statistic, in step 430. Thesystem can use the methods described herein to calculate a relationshipfor different inputted investments. For instance, the system may firstdetermine whether the investments inputted have a positive or negativeunion (co-movement). Based on the identified co-movements, the systemmay then calculate an expected return for the investments in totality.The expected return may be a time-dependent variable. As a result, thesystem may either calculate the expected return for the investments fora time period identified by the user. Alternatively, the system maygenerate an expected return for multiple time periods. For instance, thesystem may calculate and display a projected expected return in shortterm (e.g., 6 months or 1 year) and medium/long term (e.g., 5 years and10 years).

Optionally, the user can adjust the investments or an allocation ofcapital to the investments, in step 440. In response, the system canre-calculate the expected return using the Gerber relationship, in step450. As described above, the system may display how the capital isallocated to each investment and a corresponding expected return. Thesystem may provide the user the opportunity to simulate differentscenarios by allowing the user to revise the investments and/or thecapital allocated to each investment. For instance, the user may add orremove an investment to the list of investments. As a result, the systemmay re-calculate the expected return and display the results. In anotherexample, the user may revise how the capital is allocated to eachinvestment. As a result, the system may re-calculate the expected returnand display the results. The system may provide a simulation platformwhere users can run different scenarios and identify correspondingresults.

Using the methods described herein, the system may also recommend aninvestment strategy that would yield better results. The system mayexecute multiple scenarios in which different investment strategies areused. For instance, the system may determine whether a linear ornon-linear hedge should be used. The system may also determine the typeof hedge that should be used (e.g., put option). In another example, thesystem may determine which (if any) assets should be included orexcluded, such as including various environmental, social, andgovernance (ESG) investments. The system may then display the resultsgenerated by simulating different investment strategies and receive aselection from the user. Alternatively, the system may automaticallyselect a best investment strategy based on predetermined rules andcriteria (e.g., select the investment strategy that yields the bestreturn in short term or long term).

The system may use the methods and systems described herein to createcustomized analysis for different portfolios and portfolio managers. Forinstance, the system may retrieve data needed to perform the analysisand to calculate the Gerber relationship for various assets managed by aparticular portfolio manager. The system may first query a database toidentify assets being managed by a particular portfolio manager. Thesystem may then determine one or more indices associated with theportfolio manager. The system may then save the data within thepre-loaded cluster or template. A user may access a graphical userinterface hosted or generated by the system to execute the pre-loadedclusters.

Referring to FIG. 6 , when a user accesses a graphical user interface600, the system displays various preloaded (or pre-generated) clustersand templates to be executed (e.g., clusters represented by a set ofgraphical components 610). When a user selects a preloaded cluster, thesystem executes the analytical methods described herein to calculate theGerber statistic between the assets identified within the preloadedcluster (or inputted by a user). The system may then display theresults, such as by displaying any of the graphical user interfacesdiscussed herein.

The templates and clusters may account for various predeterminedstrategies for different investments and portfolios, as depicted by thecorresponding graphical component. The cluster represented by “credit”(graphical component 630) corresponds to a strategy used by all (or aportion of) portfolio managers. Using preloaded clusters, a user canview results associated with different hedge fund baskets. For instance,when the user selects graphical component 640 for the preloaded clusterof North American Long and Short (NA L/S), the system will show resultsfor a series of predetermined assets associated with the selected basketof assets.

In another example, the preloaded cluster PM1 vs. Indices (representedby graphical component 620) is customized for a particular portfoliomanager, PM1. The system may calculate a set of attributes (e.g.,indices) to be analyzed for assets managed by PM1. The preloaded clusterfor PM1 may also include the assets being managed by PM1. Additionallyor alternatively, the preloaded cluster represented by the graphicalcomponent 620 may also include relevant indices that have been selectedfor PM1 (based on various rules). The system may use various rules andcomputer models to determine an ideal set of indices for each user(e.g., each portfolio manager). For instance, the system may include S&Pindices for PM1. However, because PM1 is a merger arbitrage portfoliomanager, the system may also include indices that are specific to mergerarbitrage portfolios (e.g., indices that track the performance ofmergers) because PM1 manages assets that may have risk regardingdifferent market factors and market measures. Therefore, the preloadedcluster for PM1 may use different indices as the preloaded cluster forother portfolio managers (e.g., PM5).

In another example, the system may evaluate a PM's portfolio of returnsboth at the portfolio level and ‘sub-portfolios’ consisting of a subsetof investments in the portfolio. The system could then evaluate the listof assets that exhibit the greatest relationships with the portfolio'sreturns against the investments in the portfolio. In doing so, thesystem could help identify themes or relationships amongst investmentsin the portfolio. The system can also identify investments that arecontributing to that relationship allowing for better overall allocationof resources. The system may re-use the identified relationships orthemes by applying them to other PMs or portfolios and theircorresponding assets.

The system may periodically execute the preloaded clusters, such asdaily, weekly, or any other frequency determined by a systemadministrator. The system may have the results available, such thatdifferent authorized users can view the results by interacting with thegraphical user interface 600.

The system may also allow users to generate customized data analysisbased on their chosen criteria. As depicted in FIG. 7 , the system mayallow a user to generate any combination of data to be analyzed per userselections. Using the input elements depicted in the graphical userinterface 700, a user may create a customized way of analyzing the data.For instance, the user may select trades from the list of input elementswithin the set of graphical components 710, select gains/losses usingthe input element 720, select an index gain/loss using the input element730, and select the observation period using the input elements 740 and750. Upon generating a customized cluster, the system may analyze thedata and direct the user to FIGS. 8-9 .

Referring now to FIG. 8 , an example of a graphical user interfacedisplayed by the system is depicted. The system may use the preloadedcluster (FIG. 6 ) or customized clusters (FIG. 7 ) to analyze the data.Based on the Gerber relationships between various assets or variables(e.g., deal records), the system can display the extent of arelationship between two or more assets or the relationship of an assetto an index, as depicted in the graphical user interface 800. Therelationship can be depicted in a format whereby assets moving in thesame direction and having a positive relation may be depicteddifferently (e.g., via alphanumerical representation (e.g., numbers orclasses), different size, color, or shape) than those assets moving in adifferent direction and having a negative relation. The system mayemploy an algorithm to highlight those assets that have, for example,exhibited the most significant moves and/or have the most significantrelationships.

The grid depicted in FIG. 8 has an x-axis with a separate column foreach asset and a y-axis with a separate row for each asset. Theintersection between an asset on the x-axis and an asset on the y-axisindicates the Gerber relationship between the two assets. The Gerberrelationship can be shown as a number by applying one or more versionsof the Gerber statistic methods on the corresponding data (e.g.,performance of each asset in accordance with a particular index within adefined observation period). The system can display an indicatorrepresenting the similarity of movements across assets. For example, thesystem can provide a percentage value representing the number of periodswhere the two assets moved in the same direction minus the number ofperiods where the two assets moved in opposite directions, and thatnumber is divided by the total number of periods exceeding thethreshold, as shown in box 810 (e.g., 40%). For example, a percentage of40% may be the result of seven periods where the two deal code recordsmoved in the same direction minus three periods where the deal coderecords moved in opposite directions, divided by ten periods that exceedthe threshold criteria for that date range.

The system may also display the result in another visual format, asdepicted in FIG. 9 . As depicted, the graphical user interface 900 showsa scatter plot where different assets/variables are represented bydifferent graphical indicators (e.g., data points) separated intodifferent quadrants. The graphical user interface 900 includes fourquadrants separated by various predetermined and/or revisablethresholds. For brevity and clarity, the graphical user interface 900depicts co-movement of two assets (deal records). However, in otherembodiments, a user may customize one or more assets, such that moreassets are shown. In some configurations, the system may direct the userto the graphical user interface 900 when the user interacts with any ofthe indicators shown in FIG. 8 . For instance, when a user clicks on thebox 810, the system directs the user to the graphical user interface 900where the corresponding two assets are compared using one or moreversions of the Gerber statistic.

In FIG. 9 , a first axis 901 represents movements of a first asset(GS-HF-LS) and a second axis 902 represents movements of a second asset(EEM). A threshold value for movement of either asset may be set by thesystem and/or the user or the system administrator, which is depicted bythreshold values 903, 904, 905, and 906. These threshold values createfour quadrants: quadrant 910 (Q1) representing both assets moving in apositive direction beyond the threshold, quadrant 920 (Q2) representingthe first asset moving in a negative direction beyond the threshold andthe second asset moving in a positive direction beyond the threshold,quadrant 930 (Q3) representing both assets moving in a negativedirection beyond the threshold, and quadrant 940 (Q4) representing thefirst asset moving in a positive direction beyond the threshold and thesecond asset moving in a negative direction beyond the threshold.Quadrants 910, 930 represent the instances of a positive union, whereasquadrants 920, 940 represent the instances of a negative union. Thesystem, by default, may identify and use whatever thresholds were usedin the portfolio level analysis. However, these thresholds are notlimited to the thresholds used at the portfolio level analysis. Forinstance, an end user (PM) or a system administrator may revise thethresholds accordingly.

In the depicted embodiment, the grid uses daily measurements over anobservation period indicated by the graphical component 960 (e.g., Mar.22, 2021 to Apr. 5, 2021). The observation period may be revised by theuser. For instance, the user may instruct the system to analyze the datafor a longer period of time (e.g., 45 days) or analyze the data based onbi-weekly measurements instead of daily measurements. For each daywithin the observation period indicated within the graphical component960, a point is positioned on the grid depicted within the graphicaluser interface 900 corresponding to the movements of the two assets. Forinstance, points 911-913 and 931-934 represent co-movement of the twoassets. In contrast, points 921-923 represent a negative union (e.g.,opposite of the co-movement) of the two assets.

The system may also display the graphical component 950 where thecalculated relationship for each day is presented. In someconfigurations, the user may interact with the values depicted withinthe graphical component 950 and the system may direct the user toanother page displaying more detailed data (e.g., positions for eachasset or market movement).

Because GS1 and GS2 are less restrictive than GS, the system can analyzemore data points without excluding them due to the data points fallingbelow the restrictive thresholds. As a result, the graphical userinterface 900 does not include any data points that fall in between thethresholds 903-906.

While the embodiment shown in FIG. 9 relates to measuring movements inmonetary value with a threshold specified in dollars, it is understoodthat any suitable measurement or unit can be used for movement and anysuitable measurement or unit can be used as a threshold. For example,the movement measurement or a threshold can be absolute (e.g., a numberof units) or relative (e.g., a percentage). In some embodiments, athreshold can be a relative measurement based on past behavior of theassets. The threshold can be based upon a standard deviation of pastasset movement, whereby a lower standard deviation can represent a lowerthreshold more sensitive to asset movement. For example, a threshold maybe set to a multiple of the asset's standard deviations based on pastbehavior. In some embodiments, a threshold may be dynamically adjustedfor each measurement based on recent behavior of the asset. In suchembodiments, the threshold may automatically change over time as thebehavior of the asset evolves.

While the embodiment shown in FIG. 9 applies the same threshold value toboth assets, it is understood that a different threshold can be appliedto each asset. In some embodiments, each asset can have its ownthreshold based upon that particular asset's unique characteristics orpast performance. For example, the threshold for each asset may beselected so that it corresponds to the movement magnitude, volatility,or other historical behavior of each asset. In one embodiment, a usercan adjust the threshold for one or both assets, a feature that may beused to manually adjust for measurement sensitivity. In other examples,the system may consider the performance data in terms of a changing oflevels associated with performance of the asset, such as percentagechange (not absolute amount), log, simple difference between two assets,deviation from a trend, and the like.

In some configurations, the system may identify different benchmarks andindices to be used in the calculations discussed herein. The system maydynamically monitor performance of a certain sector or index. If theperformance satisfies a threshold, the system may generate arecommendation accordingly. For instance, if the system determines thatthe retail sector has had a sudden spike, the system may recommendcalculating a portfolio's exposure against indices corresponding to theretail sector. The system may generate an electronic notificationinforming the user (e.g., portfolio manager) that the retail sector'sperformance has had a sudden spike. The notification may then recommendusing the retail sector as a benchmark, such that a portfolio's exposureis calculated against new indices. Upon receiving authorization from theuser or a system administrator, the system may then re-analyze the datausing the updated (or additional) benchmarks.

The system may continuously monitor the market to recommend newbenchmarks, such that data is periodically calculated using updatedbenchmarks that reflect the latest market movements.

The system may also generate a confidence score for the resultscalculated. For instance, when a positive or negative union isidentified, the system may determine whether the data indicating theresult is statistically significant using another statisticalsignificance protocol (after making distributional assumptions).Specifically, the system may determine a degree of statisticalsignificance for a positive or a negative union. Statisticalsignificance indicates whether the results generated by applying one ormore versions of the Gerber statistic to the data is likely to occurrandomly (by chance) or likely to be attributable to a specific cause.If the Gerber statistic is applied to a small data sample (e.g., smallnumber of observations), it may not yield results that are statisticallysignificant. Therefore, the system may assign a low confidence score tothe result. If the system determines that the results have a lowconfidence score (e.g., a confidence score that is less than athreshold), the system may recommend increasing the observation periodto re-analyze the data using a bigger sample size. For instance, if theuser instructs the system to analyze performance data for a week (e.g.,via interacting with the input elements of the graphical component 960),the system may display a message that recommends increasing the time toa month (and/or increasing the frequency of observations to hourly)because a week (and/or daily frequency) may not yield results that arestatistically significant.

In some configurations, the system may dynamically calculate thresholdsthat would yield results with high confidence score (e.g., results thatare statistically significant). The system may vary the threshold (e.g.,thresholds that are visually depicted as lines 903, 904, 905, and 906).For instance, instead of receiving the observation period from a user,the system may automatically analyze the market based on the selectedindices and determine thresholds that would yield better results. Inthis way, the user may only select the assets to be analyzed and thesystem may automatically determine a suitable time threshold that arecustomized based on market volatility, availability of data, historicalobservations, and the like. The thresholds may also be calculated basedon the assets to be analyzed.

The system may determine the time threshold based on various attributesof the assets to be analyzed, such as price, trade volume, and the like.For example, a first stock may have more observable data points in ashorter period of time because the first stock has been traded morefrequently than a second stock. Therefore, the system may calculate adifferent observation period for the first stock than the second stock.In another example, the system may impose additional thresholds or maysegment the time windows differently based on trading price and/orvolume. For instance, the system may only analyze the data when a stockhas been traded more than a certain volume. In another example, thesystem may segment the observable periods of time into bi-daily (and notdaily) segments because a particular stock has a high trading volume.

In another example, an asset (e.g., a particular stock) may becontinuously traded during market hours. As a result, the system maycompare ownership of a stock (pricing every second) in light of thecapital invested in a strategy with a lock-up or less frequentobservable returns (e.g. hedge fund or private equity).

The system may also use different versions of the Gerber statisticdescribed herein to show multiple sets of results. While FIG. 9 depictsone set of results, the system may utilize GS, GS2, and/or GS3 togenerate different sets of results. In some configurations, thegraphical user interface 900 may include an input element (e.g., toggle,drop down menu, or a radio button) that allows the user to instruct thesystem to use a particular version of the Gerber statistic to calculatethe results. In some configurations, the system may simultaneouslydisplay two or three sets of results where each set of results iscalculated using a different version of the Gerber statistic. The systemmay also display an average of multiple Gerber statistics as the onlyset of results.

The system may also analyze the shape of the scatter plot (e.g.,arrangement and shape of the data points within each quadrant) torecommend an investment strategy. For instance, the arrangement of thedata points may indicate that when the market is in red (e.g., lowerthan a threshold), assets within a portfolio experience a decreasedvalue. However, the assets are not participating in the market when themarket is in green. Therefore, the system may recommend a new investmentstrategy (e.g., purchasing put options). In another example, if aportfolio manager has invested in stock A and shorted stock B, theportfolio is exposed to a high risk. However, based on the system'srecommendation, the portfolio manager may purchase put options insteadof shorting stock B. As a result, the portfolio's risk is limited to afixed amount. In another example, if the system determines that a linearhedge has caused noisy data, the system may recommend a non-linearhedge.

The system may retrieve one or more of the criteria, thresholds, orother data needed to generate the graphical user interface 900 from atemplate (pre-made cluster) associated with the user viewing thegraphical user interface 900 and/or a user associated with the assetsanalyzed (e.g., portfolio manager). For instance, a user may login tothe electronic platform provided by the system and select a generatedcluster. Upon instructing the system to execute the generatedcluster/template, the system may automatically retrieve the datanecessary to generate the graphical user interface 900. For instance,the system may retrieve the customized observation period thresholds,indices, and other data from the cluster/template to calculate therelationships discussed herein.

Referring now to FIG. 10 , a method 1000 depicts a method for portfolioconstruction, analysis, and visualization according to an embodiment.The method 1000 may be performed by a server, such as the server withinthe system 100 (FIG. 1 ).

At step 1010, the system may retrieve performance data for a pluralityof data records within an observation period. The system may query andretrieve performance data associated with one or more assets. Theperformance data may be filtered in accordance with various criteria,such as observation period thresholds, performance values with respectto particular indices, and the like. In some embodiment, theabove-described criteria may be retrieved from pre-generatedtemplates/clusters. For instance, based on a user identifier, the systemmay retrieve an appropriate template/cluster (e.g., a template/clustergenerated for a particular portfolio manager). In other embodiments, theabove-described criteria may be inputted by a user (e.g., FIG. 7 ).

At step 1020, the system may for at least one pair of data recordswithin the plurality of data records, determining whether a first datarecord of a pair of data records and a second data record of the pair ofdata records have a positive union or a negative union based on eachinstance in which a respective value of the performance data for eachdata record is above an upper threshold or below a lower threshold forthe first data record or the second data record. At step 1030, thesystem may display on a graphical user interface, a representation ofthe positive or negative union.

The system may apply various analytical methods discussed herein toidentify relationships between data points representing differentassets. For instance, the system may use one or more versions of theGerber statistic to populate the graphical user interface described inFIG. 8 .

At step 1040, the system may in response to receiving an indication ofinteraction with the representation of the positive or negative union,dynamically revising, by the server, the graphical user interface bydisplaying, for the pair of data records, a visual indicator within fourregions, wherein: a first region represents positive union with respectto the upper threshold and the lower threshold, a second regionrepresents negative union with respect to the upper threshold and thelower threshold, a third region represents positive union with respectto the lower threshold and negative union with respect to the upperthreshold, and a fourth region represents negative union with respect tothe lower threshold and positive union with respect to the upperthreshold.

When a user interacts with an interactive element displayed (e.g., whena user interacts with the box 810 depicted in FIG. 8 ), the system maydirect the user to a new page or may dynamically revise the graphicaluser interface. The new page or the revised graphical user interface maypresent analysis of two or more assets (e.g., a portfolio), such asdepicted in FIG. 9 .

Unless specifically stated otherwise as apparent from the followingdiscussion, it is appreciated that throughout the description,discussions utilizing terms such as “creating,” “executing,”“providing,” “calculating,” “processing,” “computing,” “transmitting,”“receiving,” “determining,” “displaying,” “identifying,” “presenting,”“establishing,” or the like, can refer to the action and processes of adata processing system, or similar electronic device, that manipulatesand transforms data represented as physical (electronic) quantitieswithin the system's registers or memories into other data similarlyrepresented as physical quantities within the system's memories orregisters or other such information storage, transmission or displaydevices. The system can be installed on a mobile device.

The embodiments can relate to an apparatus for performing one or more ofthe functions described herein. This apparatus may be speciallyconstructed for the required purposes, or it may comprise a generalpurpose computer selectively activated or reconfigured by a computerprogram stored in the computer. Such a computer program may be stored ina machine (e.g. computer) readable storage medium, such as, but notlimited to, any type of disk including floppy disks, optical disks,CD-ROMs and magnetic-optical disks, read only memories (ROMs), randomaccess memories (RAMs), erasable programmable ROMs (EPROMs),electrically erasable programmable ROMs (EEPROMs), magnetic or opticalcards, or any type of media suitable for storing electronicinstructions, and each coupled to a bus.

The embodiments described herein are described as software executed onat least one server, though it is understood that embodiments can beconfigured in other ways and retain functionality. The embodiments canbe implemented on known non-transitory devices such as a personalcomputer, a special purpose computer, cellular telephone, personaldigital assistant (“PDA”), a digital camera, a digital tablet, anelectronic gaming system, a programmed microprocessor or microcontrollerand peripheral integrated circuit element(s), an ASIC or otherintegrated circuit, a digital signal processor, a hard-wired electronicor logic circuit such as a discrete element circuit, a programmablelogic device such as a PLD, PLA, FPGA, PAL, or the like. In general, anydevice capable of implementing the processes described herein can beused to implement the systems and techniques according to thedisclosure.

It is to be appreciated that the various components of the technologycan be located at distant portions of a distributed network and/or theInternet, or within a dedicated secure, unsecured and/or encryptedsystem. Thus, it should be appreciated that the components of the systemcan be combined into one or more devices or co-located on a particularnode of a distributed network, such as a telecommunications network. Aswill be appreciated from the description, and for reasons ofcomputational efficiency, the components of the system can be arrangedat any location within a distributed network without affecting theoperation of the system. Moreover, the components can be embedded in adedicated machine.

Furthermore, it should be appreciated that the various links connectingthe elements can be wired or wireless links, or any combination thereof,or any other known or later developed element(s) that is capable ofsupplying and/or communicating data to and from the connected elements.The term module as used herein can refer to any known or later developedhardware, software, firmware, or combination thereof that is capable ofperforming the functionality associated with that element. The terms“determine,” “calculate” and “compute,” and variations thereof, as usedherein are used interchangeably and include any type of methodology,process, mathematical operation or technique.

The embodiments described above are intended to be exemplary. Oneskilled in the art recognizes that there are numerous alternativecomponents and embodiments that may be substituted for or included inthe particular examples described herein and such additions orsubstitutions still fall within the scope of the invention.

What is claimed is:
 1. A method comprising: responsive to receiving, bya server, an input of a capital amount and an acceptable risk thresholdfor a portfolio; retrieving, by the server, performance data for a setof data records within an observation period; iteratively simulating, bythe server, a plurality of allocations to at least one data record ofthe set of data records to predict an expected return value for eachallocation; for each pair of data records within the set of datarecords, determining, by the server, whether a first data record of eachpair of data records and a second data record of each pair of datarecords have a positive union or a negative union based on each instancein which a respective value of the performance data for each data recordis above an upper threshold or below a lower threshold for the firstdata record or the second data record; selecting, by the server, asubset of the set of data records, wherein each pair of data recordswithin the subset of the set of data records has a total positive unionsand negative unions that correspond to the acceptable risk threshold,and wherein the selected subset of the set of data records based on asimulated allocation having the expected return value that satisfies areturn threshold; and automatically allocating, by the server, at leasta portion of the capital amount to the subset of the set of datarecords.
 2. The method of claim 1, further comprising: periodicallymonitoring, by the server, performance data for the subset of the set ofdata records.
 3. The method of claim 1, further comprising: changing, bythe server, at least one data record within the subset of the set ofdata records with a second data record within the set of data recordsthat is not included within the subset of the set of data records. 4.The method of claim 1, further comprising: generating, by the server, anexpected rate of return for the portfolio.
 5. The method of claim 1,further comprising: retrieving, by the server, at least one of theobservation period, the upper threshold, or the lower threshold from atemplate that is customized for a user or the portfolio.
 6. The methodof claim 1, further comprising: displaying, by the server on a graphicaluser interface, a list of the subset of the set of data records.
 7. Themethod of claim 1, wherein the set of data records is received from auser.
 8. The method of claim 1, further comprising: displaying, by theserver on a graphical user interface, for at least one pair of datarecords, a visual indicator within four regions, wherein: a first regionrepresents positive union with respect to the upper threshold and thelower threshold, a second region represents negative union with respectto the upper threshold and the lower threshold, a third regionrepresents positive union with respect to the lower threshold andnegative union with respect to the upper threshold, and a fourth regionrepresents negative union with respect to the lower threshold andpositive union with respect to the upper threshold.
 9. The method ofclaim 8, wherein representation of the positive or negative union is analphanumerical representation or a graphical representation.
 10. Acomputer system comprising: a non-transitory storage medium configuredto store a set of instructions that when executed by a processor causethe processor to: responsive to receive an input of a capital amount andan acceptable risk threshold for a portfolio; retrieve performance datafor a set of data records within an observation period; iterativelysimulate a plurality of allocations to at least one data record of theset of data records to predict an expected return value for eachallocation; for each pair of data records within the set of datarecords, determine whether a first data record of each pair of datarecords and a second data record of each pair of data records have apositive union or a negative union based on each instance in which arespective value of the performance data for each data record is abovean upper threshold or below a lower threshold for the first data recordor the second data record; select a subset of the set of data records,wherein each pair of data records within the subset of the set of datarecords has a total positive unions and negative unions that correspondto the acceptable risk threshold, and wherein the selected subset of theset of data records based on a simulated allocation having the expectedreturn value that satisfies a return threshold; and automaticallyallocate at least a portion of the capital amount to the subset of theset of data records.
 11. The system of claim 10, wherein the set ofinstructions further cause the processor to: periodically monitorperformance data for the subset of the set of data records.
 12. Thesystem of claim 10, wherein the set of instructions further cause theprocessor to: change at least one data record within the subset of theset of data records with a second data record within the set of datarecords that is not included within the subset of the set of datarecords.
 13. The system of claim 10, wherein the set of instructionsfurther cause the processor to: generate an expected rate of return forthe portfolio.
 14. The system of claim 10, wherein the set ofinstructions further cause the processor to: retrieve at least one ofthe observation period, the upper threshold, or the lower threshold froma template that is customized for a user or the portfolio.
 15. Thesystem of claim 10, wherein the set of instructions further cause theprocessor to: display, on a graphical user interface, a list of thesubset of the set of data records.
 16. The system of claim 10, whereinthe set of data records is received from a user.
 17. The system of claim10, wherein the set of instructions further cause the processor to:display on a graphical user interface, for at least one pair of datarecords, a visual indicator within four regions, wherein: a first regionrepresents positive union with respect to the upper threshold and thelower threshold, a second region represents negative union with respectto the upper threshold and the lower threshold, a third regionrepresents positive union with respect to the lower threshold andnegative union with respect to the upper threshold, and a fourth regionrepresents negative union with respect to the lower threshold andpositive union with respect to the upper threshold.
 18. The system ofclaim 17, wherein representation of the positive or negative union is analphanumerical representation or a graphical representation.
 19. Asystem comprising: a server configured to: responsive to receive aninput of a capital amount and an acceptable risk threshold for aportfolio; retrieve performance data for a set of data records within anobservation period; iteratively simulate a plurality of allocations toat least one data record of the set of data records to predict anexpected return value for each allocation; for each pair of data recordswithin the set of data records, determine whether a first data record ofeach pair of data records and a second data record of each pair of datarecords have a positive union or a negative union based on each instancein which a respective value of the performance data for each data recordis above an upper threshold or below a lower threshold for the firstdata record or the second data record; select a subset of the set ofdata records, wherein each pair of data records within the subset of theset of data records has a total positive unions and negative unions thatcorrespond to the acceptable risk threshold, and wherein the selectedsubset of the set of data records based on a simulated allocation havingthe expected return value that satisfies a return threshold; andautomatically allocate at least a portion of the capital amount to thesubset of the set of data records.
 20. The system of claim 19, whereinthe server is further configured to periodically monitor performancedata for the subset of the set of data records.